We discuss the dynamical properties of a SIRS computer virus propagation model with impulsive detoxication and saturation effect. By the Internet, new antivirus software can be released immediately and take effect quickly after it is running. This leads to the circumstance that many infected computers can be cured in a short time. So impulsive detoxication is a vitally important way for prohibiting the spread of network viruses. The theoretical results show that: (a) the virus-free equilibrium is globally stable when the basic reproduction ratio (BRR) is less than unity, (b) the system is uniformly permanent when BRR exceeds unity, and (c) a supercritical bifurcation occurs when BRR equals unity. Several numerical examples also clearly display the results obtained. Finally, some feasible strategies of eradicating electronic viruses are advised.

While the popularized communication networks have brought great convenience to our daily work and life, they also provide a fast channel for the spread of computer viruses. During the last decades, massive outbreaks of network viruses have caused enormous financial losses.

For the purpose of effectively controlling virus diffusion, it is of practical importance to understand the way that malicious codes propagate over the Internet. Due to the compelling analogy between electronic viruses and their biological counterparts [1, 2], a multitude of computer virus epidemic models, ranging from SIS models [3, 4], SIR models [5, 6], SIRS models [7], SEIR models [8], SEIRS model [9], SLBS models [10–12], SLAS model [13], SIPS model [14], to delayed models [7, 15] and stochastic models [16, 17], have been proposed. In our opinion, these models can help us better understand how the viruses diffuse on networks.

In the classical epidemic models, the incidence rate β, which is the probability of transmission per contact, is assumed to be bilinear with respect to the numbers of susceptible and infected individuals [18]. To the best of our knowledge, in most of previous computer virus models, the incidence rate β is also simply supposed to be unvaried [19, 20]. However, in reality, when the population are aware of the existence of computer viruses on networks, they will reduce the number of communication contacts per unit time to avoid being infected, the more infectious computers being reported, the less contact with other computers, which is called the saturation effect. In an attempt to capture the saturation effect, Yuan et al. [21] proposed a nonlinear incidence rate. As the transmission details of computer infections are quite complicated, their results have only limited applications. Recently, Yang and Yang [22] also suggested a nonlinear incidence rate. However, its description of saturation effect is not clear. Based on a large number of numerical examples, we use \(1/(1+\alpha I(t))\) (α is a positive constant and \(I(t)\) represents the percentage of infected internal computers at time t, respectively) to denote the saturation effect [23], namely, use \(\frac{\beta}{1+\alpha I(t)}\) to replace β. This seems reasonable.

In the prevalence of infectious diseases, it is impossible to rapidly disseminate the new remedy to a great many patients, and it often takes a lot of time to finish one or several courses of the remedy [24–26]. On the contrary, as new antivirus software can be released immediately via the Internet and takes effect quickly after it started running, many of the infected computers can be cured in a short time [20]. To understand how impulsive detoxication [16] and saturation effects prohibit virus spread on networks, a novel impulsive computer virus propagation model is established in this paper. We theoretically analyze that the virus-free equilibrium is globally asymptotically stable when BRR is less than unity. It is shown that the system is uniformly permanent when BRR exceeds unity. In addition, by bifurcation theory we see that a supercritical bifurcation occurs when BRR equals unity. Both theoretical predictions and numerical examples show that impulsive detoxication can control virus diffusion effectively. Finally, considering the influence of varying model parameters on BRR, we propose some feasible strategies of deracinating malicious viruses. One should emphasize that there are few computer virus propagation models in the literature that consider the combined impact of impulsive detoxication and saturation effects.

The remainder of this paper is organized as follows: the new model is elaborated in Section 2; it is theoretically studied in Sections 3, 4, and 5; the influence of model parameters is discussed in Section 6; finally, this work is summarized in Section 7.

In this section, we will formulate a computer virus propagation model with impulsive detoxication and saturation effects. As usual, we shall simply assume that every computer is in one of three states: susceptible, infected, and recovered. Both susceptible computers and recovered computers are uninfected, however, the former has no immunity, while the latter has temporary immunity. Let \(S(t)\), \(I(t)\), and \(R(t)\) denote the percentages of susceptible, infected, and recovered internal computers at time t, respectively, then \(S(t)+I(t)+R(t)\equiv1\).

For simplicity, in this paper, we do not consider the situation where the susceptible computers get recovered.

Considering the above discussions, we establish a mathematical model with the following assumptions:

(H1)

All newly accessed computer are susceptible. Furthermore, susceptible computers are accessed to the Internet at the constant rate \(\mu>0\). And every computer on the Internet leaves the network with constant probability \(\mu>0\).

(H2)

By contact with infected internal computers, at time t, every susceptible internal computer gets infected with probability \(\frac{\beta I(t)}{1+\alpha I(t)}\), where \(\beta>0\) and \(\alpha>0\) are constants.

In this section, we shall prove the existence and global stability of the infection-free equilibrium under certain conditions.

First, we show the existence of the infection-free equilibrium, in which infectious computers are entirely absent from the internal computers permanently, i.e., \(I(t)\equiv0\), \(t\geq0\). In this situation, the growth of the removed computers \(R(t)\) simplifies to

with initial condition \(v(0^{+})=I(0^{+})\). Then \(v(T^{+})=I(0^{+})\sigma\), \(v(nT^{+})=I(0^{+})\sigma^{n}\), which implies that \(\lim_{n\rightarrow\infty }v(nT^{+})=0\). Moreover, if \(nT< t\leq(n+1)T\), then

implying that \(\lim_{t\rightarrow+\infty}v(t)=0\). By the comparison theorem in impulsive differential equations [27], we get \(\lim_{t\rightarrow+\infty}I(t)=0\). Thus, there is \(T_{1}>0\) such that \(I(t)<\varepsilon\) for \(t\geq T_{1}\). Substituting this equation into system (2), we have

From Figures 1 and 2, we can see that the state of system (1) is approaching the virus-free equilibrium, as fits the theoretical prediction. In other words, the computer viruses will be eradicated in these situations.

In the following, we study uniform permanence of system (2) and system (1). First we give two definitions.

Definition 1

System (2) is said to be virus permanent in Ω, if for every solution \((I (t), R(t))\) to system (2) with \(I (0^{+} ) > 0\), there is \(m > 0\) such that \(I (t)\geq m\) for all large t.

Definition 2

System (2) is said to be uniformly permanent in Ω, if there is a constant \(c> 0\) (independent of the initial data), such that every solution \((I(t),R(t))\) with initial condition \((I(0^{+}),R(0^{+})) \in\Omega\) of system (2) satisfies

Hence, \(I(kT)\geq I(N_{2}T)\sigma^{k-N_{2}}\) for \(k\geq N_{2}\). Noting that \(I(N_{2}T)>0\), we have \(\lim_{k\rightarrow+\infty}I(kT)=+\infty\). It contradicts the fact that \(I(t)\leq1\). So there exists a \(t_{2}>t_{1}\), such that \(I(t_{2})\geq m_{1}\).

If \(I(t)\geq m_{1}\) for all \(t\geq t_{2}\), the claimed result has been obtained. Now, assume \(I(t) < m_{1}\) for some \(t>t_{2}\). Let

$$t_{3}=\inf_{t>t_{2}}\bigl\{ t:I(t)< m_{1}\bigr\} . $$

There are two cases here.

Case 1. \(t_{3}= kT\), without loss of generality, let \(t_{3}=K_{1}T\) (\(K_{1}\) is a positive integer). Then \(I(t)\geq m_{1}\) for \(t\in [t_{2},t_{3})\), and \(I(t)\) is continuous, so we have \(I(t_{3})=m_{1}\), and \(I(t_{3}^{+})=(1-q)I(t_{3})< m_{1}\). By induction, we claim that there exists a \(t_{4}\in(K_{1}T,(K_{1}+1)T]\), such that \(I(t_{4})\geq m_{1}\), otherwise, \(I(t)< m_{1}\) for \(t\in (K_{1}T,(K_{1}+1)T]\), evidently, (17) and (18) holds on this interval. So we get \(I((K_{1}+1)T)\geq I(K_{1}T)\sigma>m_{1}\), which is a contradiction. Let

$$t_{4}=\inf_{t>t_{3}}\bigl\{ t:I(t)\geq m_{1} \bigr\} . $$

Then \(t_{4}\in(K_{1}T,(K_{1}+1)T]\), \(I(t_{4})=m_{1}\), and \(I(t)< m_{1}\) for \(t\in(t_{3},t_{4})\).

By induction, we claim that there is a \(t_{5}\in(t_{4},(K_{2}+K_{3}+1)T]\), such that \(I(t_{5})\geq m_{1}\), otherwise, \(I(t)< m_{1}\) for \(t\in (t_{4},(K_{2}+K_{3}+1)T]\), obviously, (17) and (18) hold on this interval. So we have

Corollary 2

From the first equation and the fourth equation of system (1) and using Corollary 2, we have \(\frac{dS(t)}{dt} >\mu+ \delta m_{4} - (\mu+ \beta) S(t)\), \(t\geq t_{6}\); furthermore, we get \(\lim_{t\rightarrow\infty}S(t)\geq\frac{\mu+ \delta m_{4}}{\mu+ \beta}\). For a sufficiently small \(\varepsilon_{3}>0\), let \(m_{5}=\frac{\mu+ \delta m_{4}}{\mu+ \beta }-\varepsilon_{3}>0\), there exists a \(t^{\prime}>t_{6}\), such that \(S(t)>m_{5}\) for \(t>t^{\prime}\).

By combining Corollary 2 and the above discussions, we get the following.

Theorem 8

Example 3

For system (1) with the parameter values \((\mu ,\beta,\gamma,\delta,\alpha,q,T) =(0.08, 0.23, 0.06, 0.1, 0.1, 0.15, 5)\), we have \(\Re_{0}=1.3333>1\). Figures 3-6 demonstrate the time plots of \(S(t)\), \(I (t)\), and \(R(t)\) and the phase portrait for the system with initial condition \((S(0), I (0), R(0)) = (0.5, 0.2, 0.3)\), respectively. We can see without difficulty that it conforms to the theoretical prediction. In other words, the computer viruses cannot be eradicated in this situation.

Example 4

Consider system (1) with the parameter values \((\mu ,\beta,\gamma,\delta,\alpha, q,T)=(0.06, 0.28, 0.04, 0.12, 0.05, 0.15, 10)\). Then \(\Re_{0}=2.4086>1\). Figures 7 and 8 display the time plots of \(I(t)\) and the phase portrait for the system with initial condition \((S(0), I (0), R(0)) = (0.5, 0.2, 0.3)\), respectively, again it meets the theoretical prediction. In fact, the viral prevalence is quite high in this situation.

has a stable T-periodic solution denoted \(x_{e}(t)\). Thus, \(\zeta (t)=(x_{e}(t),0)^{T}\) is a trivial periodic solution to system (23). We will use bifurcation theory [29] to discuss the existence of viral periodic solution to system (2). Let \(\Phi(t)=(\phi_{1}(t),\phi_{2}(t))\) be the flow associated with system (23). We get \(X(t)=(x_{1}(t),x_{2}(t))^{T}=\Phi(t,X_{0})\), \(0< t\leq T\), where \(X_{0}=X(0)\).

Theorem 9

For system (2), a supercritical bifurcation occurs at\(\Re_{0}=1\). System (2) has a stable viral periodic solution which bifurcates from the virus-free equilibrium when\(\Re_{0}\)increasingly goes across unity.

By the previous discussions, some practicable and effective strategies should be followed to lower \(\Re_{0}\) below unity for deracinating network viruses. For that purpose, in the following, we study the effects of the changing model parameters on \(\Re_{0}\).

So we must manipulate T and q to meet (25) for eradicating computer viruses in this situation.

Remark 2

We notice that δ and α are not in (4), yet we know that they have significant impact on the spread of computer viruses. It ought to be considered as a demerit of the model, and we shall improve the model in the next research.

On the basis of the previous analysis, some practical and effective measures for containing the virus prevalence is presented below.

(1)

Strengthening the research of antivirus software and shortening the development cycle are conducive to reduce the impulsive detoxication period T and to enhance the impulsive detoxication rate q and the recovery rate γ.

(2)

The people must be often reminded to install antivirus software, so that the impulsive detoxication rate q and the recovery rate γ are raised.

(3)

Let your computer leave the Internet when unnecessary, so that the recruitment rate μ is increased, the incidence rate β is minimized.

Taking into account impulsive detoxication and saturation effects in the conventional SIRS model, a novel computer viruses propagation model has been established. The dynamical properties of this model have been investigated theoretically, and the results obtained have also been demonstrated by several numerical examples. Based on an analysis of the impact of varying model parameters on BRR, some effective measures for controlling electronic virus diffusion have been advised.

Recently, Yang et al. [30, 31] analyzed the impact of the structure of the propagation network on the spread of computer virus, and they proposed a node-based epidemic model, which can help us better understand how the electronic viruses diffuse on networks. The final goal is not only to understand epidemic processes and predict their behavior, but also to control their dynamics [32]. Our future work is to research the impact of impulsive detoxication on the spread of computer viruses in different structures of the propagation network.

Acknowledgements

This research is supported by the Natural Science Foundation of China (No. 61374078), NPRP grant # NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation), Science and technology fund of Guizhou Province (LH [2015] No. 7612), China, and Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2015jcyjBX0052), China.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

XZ conceived and designed the experiments. XZ performed the experiments. XZ, CL, and TH wrote the paper. All authors read and approved the final manuscript.